Bell's Theorem Problem

I am refering to Introduction to Quantum Mechanics (2nd Edition) by David J. Griffiths, page 425 on Bell's Theorem.

Griffiths used a parameter, called [tex]\rho(\lambda)[/tex] as the probability density for the hidden variable.

What I don't understand is that the hidden variable was suppose to make the theory deterministic, or specifically to show that quantum mechanics as an indeterministic theory is incomplete.
What is the reason that he can use probability to describe the hidden variable? Isn't this a contradiction?

This test of QM involves making many measurements on systems that have been "identically prepared". The idea of HV theory is that there might be some unknown property of the system (labeled by [itex]\lambda[/itex]) that determines exactly what will happen in that particular system, and that [itex]\lambda[/itex] varies from system to system. Thus the supposedly "identical preparation" of different systems is actually not identical, but depends in some way on the precise details and prior history of each system. This gives a statistical distribution for [itex]\lambda[/itex] which is called [itex]\rho(\lambda)[/itex].